The Tolls of Elsinore

Elsinore, in northern Denmark, was once one of the wealthiest cities of Europe. Its wealth derived from its location and a subtle threat of force. You see, its cannons dominated the narrow body of water called the Oresund. Any ship of size passing from the Atlantic into the Baltic had to pass the line of sight of those cannons. Elsinore took advantage of this strategic position, starting in 1429, by charging a toll called the Sound Dues. In the early years, the toll was the same for all ships, but in 1567 the toll was charged based on the value of the cargo on a ship.

But who determined the value? The Danes came up with a brilliant idea. If the King of Denmark decided the declaredvaluation of some ship's cargo was too low, he had the right to buy the cargo for that valuation. This rule discouraged ships from setting obviously low values on what they carried.One frequent cargo was herring. The fish would swarm at certain seasons and provided a substantial portion of the protein of Europe.

So much is historical fact. We now enter into fiction.

A group of herring merchants determined that Elsinore itself could store only a certain amount of herring or, as Shakespeare reminds us, something would beveryrotten in the state of Denmark. In fact the amount that could be stored was exactly what could be sent in one shipment.

Suppose that the toll is 25 percent (in fact the Danes normally charged far less, but our fictitious King felt strapped for cash). If a ship owner claimed that a shipment was worth 1,000 gold pieces, then the toll would be 250 gold pieces if that valuation was accepted. Otherwise the King's inspectors could decide to purchase the shipment for 1,000 gold pieces. The inspectors made their decisions quickly, so ships never spent more than a day at Elsinore.

Warm-up:

Suppose the shipper sends in one ship of herring having a sale price of 1,000 gold pieces. What value should he assert his cargo has when he speaks to the inspectors at Elsinore? His goal is to maximize the net sale price he finally receives. Assume that the inspectors know the true value.

Solution to Warm-Up:

He should claim the value is 800 gold pieces. If the inspectors decide to buy, he receives 800 gold pieces. If that valuation is accepted, then he pays 200 gold pieces in toll but sells the cargo for 1,000 gold pieces and so finishes by receiving a net 800 gold pieces. If he sets the value higher, then the inspectors will accept the toll (which will be more than 200 gold pieces) and the net sale price will be less than 800 gold pieces. If he sets the value lower, then the inspectors will buy the cargo for less than 800 gold pieces.

Problems:

Two herring shippers decide to get together to try to maximize their collective net sale prices. Their plan is to send in two ships, one per day. Each would have 1,000 gold pieces worth of herring. The claimed value of the shipment would depend on the order of the ship in the series of shipments and on whether the inspectors had already bought the contents of a previous ship. Once the inspectors bought the herring from a previous ship, all future ships would declare a value of zero because they would know that Elsinore did not have the warehouse space to store a second shipment of herring.The inspectors might grumble, but the tradition of "take the toll and let the ship go or buy the shipment" was a contractual right, binding both the shippers and the king.

1. Suppose that there were just two ships and that they arrived one day apart. Both cargoes are worth 1,000 gold pieces. What value should the first shipper declare his cargo to be worth? (Remember that the two shippers will share their net proceeds equally.)

The King, upon hearing of these low valuations, decides to build a second warehouse that can again store one shipment. In response, the shippers recruit three of their friends. They schedule all five of their ships (each holding 1,000 gold pieces worth of cargo) to arrive at the same time. The warehouse can fit the shipments of only two boats.

2. What should the shippers set their cargo values to be to maximize their overall net sale prices?

Later, the shippers get together for a beerfest in Luebeck and one asks, "Would be have been better off to send the ships in one at a time?"

3. What do you think? Hard question: How much better?

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ABOUT THE AUTHOR(S)

Dennis E. Shasha, Courant Institute of Mathematical Sciences, New York University.
Dennis's most recent puzzle book, Puzzles for Programmers and Pros, was published this past May by John Wiley and Sons/Wrox

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